Mesoscopic theory of wavefront shaping to focus waves deep inside disordered media

We describe the theory of focusing waves to a predefined focal point deep inside a disordered three-dimensional (3D) opaque medium by the shaping of $N$ different field sources outside the medium, also known as wavefront shaping. Our study is motivated by recent experiments where wavefront shaping, mediated by the controlled interference of scattered waves, coexists with a classical diffusion picture where wave interference appears irrelevant. We derive the energy density of the wave field both near the focal point and anywhere else inside the medium, averaged over realizations after optimizing the focus. To this end, we conceive of a point source at the focal point inside that emits waves to an external detector array that---by time reversal---becomes a source array that emits the desired shaped fields like a spatial light modulator. In this picture, the average energy density including the focusing is described by the ${C}_{1}, {C}_{2}, {C}_{3}$, and even ${C}_{0}$ intensity correlations known from mesoscopic transport theory. We draw the remarkable conclusion that by averaging over optimized wavefronts of the array, associated with different realizations of the disorder, a background energy density is obtained that obeys a diffusion equation in which ${C}_{2}$ correlations create an energy source inside. Hence a classical property (diffusion) coexists with interference (${C}_{2}$ correlations). We derive the peak and background energy density distributions for ${C}_{0}, {C}_{1}$, and ${C}_{2}$ correlations, including for a slab. We find that the relative enhancement of the energy density at the focal point is proportional to the angular opening of the array, which de facto replaces the role of the inverse dimensionless conductance $1/g$ as the universal parameter in the transmission statistics, and it generalizes the number of segments known from experiments. We exploit the concept of a source for ``optimized background energy'' inside the medium to compare several different energy density profiles proposed in the literature, which are associated with optimized transmission by a slab using wavefront shaping. We explain why the best matching models have the first diffusion mode dominating, as was found in earlier work. Our approach avoids random-matrix theory restricted to waveguide geometries, confirms the role non-Gaussian correlations in the wavefront shaping for focusing deep inside 3D random media, and highlights the notion of an energy source created in this process. Our results are relevant for applications where the internal energy density in opaque media is crucial, varying from white-light illumination, projection optics, semiconductor metrology, (bio)sensing, to photovoltaics.